I recently read a charming little paper: Quasi-elliptic integrals and periodic continued fractions, by van der Poorten and Tran. Most of us who have taken a number theory course of some kind learned how to solve Pell’s equation: where is a nonsquare positive integer. The usual method is to compute the continued fraction
One then defines the convergents of by
Then tends to be very small and, if you compute long enough, for some you will have .
What van der Poorten and Tran do is to ask what happens if is not an integer, but a polynomial . Before I get into details, I want to tell you about something gorgeous that I won’t explain at all. Using the methods in their paper, van der Poorten and Trap can discover identities like
Isn’t that pretty?
It turns out that the continued fraction algorithm for is actually much prettier than for integers. Everything should be understood in terms of the curve cut out by . This is a curve of genus , with two points at infinity. (One of these points is the limit of and the other is the limit of .) I’ll call these two points and . The theory is controlled by the line bundles . In particular, there are nontrivial solutions to if and only if the continued fraction is periodic, if and only if for some .
Below the fold, I’ll explain what is meant by the continued fraction algorithm for an algebraic function, and tell you some of the other nice results from the paper.
Given any power series in , we define the continued fraction of .
Define . Set and define by . Then set and . Continuing in this way, we get a sequence of polynomials, a sequence of power series, and a continued fraction
We can also define the convergents as before; they do converge to in the sense that each ratio agrees with to a higher order than the ratio does.
In particular, suppose that is a polynomial of the form .
Then is a power series in :
So we can define the continued fraction of .
We keep the notations , , and from above.
I’ll explain just one key idea from the paper. Let’s think about the zeroes and poles of . Since is a polynomial in , its only poles are at , and it has a pole of the same order at both ‘s. So, other than , the function has the same poles as . Then has zeroes at the poles of .
That’s what happens away from . Suppose that has a pole of order at , and a zero of order at . Then has a pole of order at both ‘s. The difference has a zero of order at and a pole of order at . So has a pole of order at and a zero of order at .
Summing up the last two paragraphs, let the poles of be and let the zeroes be . Then the poles of are , for some and some and the zeroes are . (Here , and are supported away from .) In other words, there is a sequence of positive integers and a sequence of divisors such that the poles of are while the zeroes are .
Note that is the degree of . Note also that in the Picard group, so
It’s not too hard to work out what happens if the coefficients of are chosen generically. The first has degree and all the other are . A bit of effort checks that has degree (exercise!), so all of have degree and, in fact, in the Picard group. You may remember that a generic divisor of degree has a unique effective representative in PIcard; is that unique representative. So, we have just found an explicit way to write down an arithmetic progression in , where is a hyperelliptic curve.
Of course, the fun comes in the nongeneric case. In that case, the can skip around. It’s really fun when is torsion in the Picard group or, in other words, when there is a unit in the coordinate ring of . Then, eventually, the sequence in Picard will repeat. It turns out, when this happens, the corresponding approximation gives your unit!
There are plenty of other ideas in the paper. What is the analogue of the result that the are the best approximations to ? The are all of the form : how do we relate the polynomials and to the divisors ? And how did I come up with that integral above? All this and more, so read the paper!